Funktionsanalyse og transformation

This monograph explores the concept of the Brouwer degree and its continuing impact on the development of important areas of nonlinear analysis. The authors define the degree using an analytical approach proposed by Heinz in 1959 and further developed by Mawhin in 2004, linking it to the Kronecker index and employing the language of differential forms. The chapters are organized so that they can be approached in various ways depending on the interests of the reader. Unifying this structure is the central role the Brouwer degree plays in nonlinear analysis, which is illustrated with existence, surjectivity, and fixed point theorems for nonlinear mappings. Special attention is paid to the computation of the degree, as well as to the wide array of applications, such as linking, differential and partial differential equations, difference equations, variational and hemivariational inequalities, game theory, and mechanics. Each chapter features bibliographic and historical notes, and the final chapter examines the full history. Brouwer Degree will serve as an authoritative reference on the topic and will be of interest to professional mathematicians, researchers, and graduate students.

In diesem Lehrbuch werden die Methoden der Funktionalanalysis mit ihren Anwendungen in der Theorie elliptischer Differentialgleichungen behandelt. Gleichzeitig werden dem Leser die analytischen und funktionalanalytischen Satze naher gebracht, die fur die numerische Approximation elliptischer (und anderer) Differentialgleichungen bedeutsam sind. Neben dem klassischen Stoff der linearen Funktionalanalysis werden daher ausfuhrlich die Sobolevschen Funktionenraume (auch von negativer und gebrochener Ordnung) sowie die Existenz- und Regularitatstheorie elliptischer Differentialgleichungen behandelt. Besonderer Wert wird auf die Umsetzung der Funktionalanalysis gelegt, also der Anwendung der abstrakten Theorie auf den konkreten Fall. Dies geschieht durch eine Vielzahl von Anwendungsbeispielen. Zahlreiche sorgfaltig ausgewahlte und kommentierte Aufgaben runden die Darstellung ab.

This textbook provides a graduate-level introduction to the spectral theory of linear operators on Banach and Hilbert spaces, guiding readers through key components of spectral theory and its applications in quantum physics. Based on their extensive teaching experience, the authors present topics in a progressive manner so that each chapter builds on the ones preceding. Researchers and students alike will also appreciate the exploration of more advanced applications and research perspectives presented near the end of the book.Beginning with a brief introduction to the relationship between spectral theory and quantum physics, the authors go on to explore unbounded operators, analyzing closed, adjoint, and self-adjoint operators. Next, the spectrum of a closed operator is defined and the fundamental properties of Fredholm operators are introduced. The authors then develop the Grushin method to execute the spectral analysis of compact operators. The chapters that follow are devoted to examining Hille-Yoshida and Stone theorems, the spectral analysis of self-adjoint operators, and trace-class and Hilbert-Schmidt operators. The final chapter opens the discussion to several selected applications. Throughout this textbook, detailed proofs are given, and the statements are illustrated by a number of well-chosen examples. At the end, an appendix about foundational functional analysis theorems is provided to help the uninitiated reader.A Guide to Spectral Theory: Applications and Exercises is intended for graduate students taking an introductory course in spectral theory or operator theory. A background in linear functional analysis and partial differential equations is assumed; basic knowledge of bounded linear operators is useful but not required. PhD students and researchers will also find this volume to be of interest, particularly the research directions provided in later chapters.

Wichtige in der Quantenmechanik auftretende Begriffe mathematisch präzise und ausführlich zu erklären und anzuwenden ¿ das ist das Ziel des vorliegenden Buches. Die Axiome der Quantenmechanik können in wenigen Zeilen formuliert werden, stecken aber voller mathematisch anspruchsvoller Begriffe. In diesem Buch werden die wichtigsten Konzepte erläutert, welche zum Verständnis der Quantenmechanik benötigt werden. Das Buch sammelt die benötigten Definitionen und Sätze aus verschiedenen Bereichen der Mathematik (unter anderem Maßtheorie, Fourieranalysis, Funktionalanalysis und Operatortheorie), wobei die Aussagen vollständig bewiesen oder mit genauen Literaturangaben belegt werden. Nachdem die mathematischen Grundlagen bereitgestellt wurden, können viele zentrale Ergebnisse der Quantenmechanik einfach gewonnen werden ¿ so besteht etwa der Beweis der Heisenbergschen Unschärferelation nur aus wenigen Zeilen. Darüber hinaus werden in diesem Buch grundlegende quantenmechanische Systeme untersucht, insbesondere wird das Spektrum des Wasserstoffatoms mit und ohne Spin vollständig hergeleitet. Durch die präzise Formulierung und die ausgeführten Beweise schließt dieses Buch eine Lücke für Studierende der Physik und Mathematik: Es setzt kein Vorwissen voraus, das über die Grundvorlesungen und Kenntnisse der ersten drei Semester hinausgeht ¿ und eignet sich damit in beiden Fächern ausgezeichnet für die zweite Hälfte des Bachelor-Studiums oder als Ergänzung im Masterbereich. Wer die Quantenmechanik bereits aus der Physik kennt, wird hier die gehörten Begriffe präzisiert und vertieft finden, und wem einige der verwendeten Theorien bereits aus dem Mathematik-Studium vertraut sind, der wird hier die Anwendung in der Quantenmechanik kennenlernen.

This book on recent research in noncommutative harmonic analysis treats the Lp boundedness of Riesz transforms associated with Markovian semigroups of either Fourier multipliers on non-abelian groups or Schur multipliers. The detailed study of these objects is then continued with a proof of the boundedness of the holomorphic functional calculus for Hodge-Dirac operators, thereby answering a question of Junge, Mei and Parcet, and presenting a new functional analytic approach which makes it possible to further explore the connection with noncommutative geometry. These Lp operations are then shown to yield new examples of quantum compact metric spaces and spectral triples.  The theory described in this book has at its foundation one of the great discoveries in analysis of the twentieth century: the continuity of the Hilbert and Riesz transforms on Lp. In the works of Lust-Piquard (1998) and Junge, Mei and Parcet (2018), it became apparent that these Lp operations can be formulated on Lp spaces associated with groups. Continuing these lines of research, the book provides a self-contained introduction to the requisite noncommutative background. Covering an active and exciting topic which has numerous connections with recent developments in noncommutative harmonic analysis, the book will be of interest both to experts in no-commutative Lp spaces and analysts interested in the construction of Riesz transforms and Hodge-Dirac operators.

Master's Thesis from the year 2015 in the subject Mathematics - Analysis, grade: A, , course: MSC Pure mathematics, language: English, abstract: In this study, the author has investigated the absolutely continuous spectrum of a fourth order self-adjoint extension operator of minimal operator generated by difference equation defined on a weighted Hilbert space with the weight function w(t) > 0, t ¿ N where p(t), q(t), r(t) and m(t) are real-valued functions. The author has applied the M-matrix theory as developed in Hinton and Shaw in order to compute the spectral multiplicity and the location of the absolutely continuous spectrum of self-adjoint extension operator. These results have been an extension of some known spectral results of fourth order differential operators to difference setting. Similarly, they have extended results found in Jacobi matrices.In this thesis, chapter 1 is about introduction and some preliminary results including literature review, objectives, methodology and basic definitions. In chapter 2, the author has given the results on the computation of the eigenvalues, dichotomy conditions and some results on singular continuous spectrum. Chapter 3 contains the main results in deficiency indices, absolutely continuous spectrum and the spectral multiplicity. Finally, the author has summarized his results in chapter 4 and also highlighted areas of further research.

The aim of this work is to initiate a systematic study of those properties of Banach space complexes that are stable under certain perturbations. A Banach space complex is essentially an object of the form 1 op-l oP +1 ... --+ XP- --+ XP --+ XP --+ ... , where p runs a finite or infiniteinterval ofintegers, XP are Banach spaces, and oP : Xp ..... Xp+1 are continuous linear operators such that OPOp-1 = 0 for all indices p. In particular, every continuous linear operator S : X ..... Y, where X, Yare Banach spaces, may be regarded as a complex: O ..... X ~ Y ..... O. The already existing Fredholm theory for linear operators suggested the possibility to extend its concepts and methods to the study of Banach space complexes. The basic stability properties valid for (semi-) Fredholm operators have their counterparts in the more general context of Banach space complexes. We have in mind especially the stability of the index (i.e., the extended Euler characteristic) under small or compact perturbations, but other related stability results can also be successfully extended. Banach (or Hilbert) space complexes have penetrated the functional analysis from at least two apparently disjoint directions. A first direction is related to the multivariable spectral theory in the sense of J. L.

During the past decade, the mathematics of superconductivity has been the subject of intense activity. This book examines in detail the nonlinear Ginzburg–Landau functional, the model most commonly used in the study of superconductivity. Specifically covered are cases in the presence of a strong magnetic field and with a sufficiently large Ginzburg–Landau parameter kappa. Key topics and features of the work: * Provides a concrete introduction to techniques in spectral theory and partial differential equations * Offers a complete analysis of the two-dimensional Ginzburg–Landau functional with large kappa in the presence of a magnetic field * Treats the three-dimensional case thoroughly * Includes open problems Spectral Methods in Surface Superconductivity is intended for students and researchers with a graduate-level understanding of functional analysis, spectral theory, and the analysis of partial differential equations. The book also includes an overview of all nonstandard material as well as important semi-classical techniques in spectral theory that are involved in the nonlinear study of superconductivity.

In contrast to other books devoted to the averaging method and the method of integral manifolds, in the present book we study oscillation systems with many varying frequencies. In the process of evolution, systems of this type can pass from one resonance state into another. This fact considerably complicates the investigation of nonlinear oscillations.In the present monograph, a new approach based on exact uniform estimates of oscillation integrals is proposed. On the basis of this approach, numerous completely new results on the justification of the averaging method and its applications are obtained and the integral manifolds of resonance oscillation systems are studied.This book is intended for a wide circle of research workers, experts, and engineers interested in oscillation processes, as well as for students and post-graduate students specialized in ordinary differential equations.

This book consists of a collection of original, refereed research and expository articles on elliptic aspects of geometric analysis on manifolds, including singular, foliated and non-commutative spaces. The topics covered include the index of operators, torsion invariants, K-theory of operator algebras and L2-invariants. There are contributions from leading specialists, and the book maintains a reasonable balance between research, expository and mixed papers.

The many diverse articles presented in these three volumes, collected on the occasion of Alexander Grothendieck's sixtieth birthday and originally published in 1990, were offered as a tribute to one of the world's greatest living mathematicians. Grothendieck changed the very way we think about many branches of mathematics. Many of his ideas, revolutionary when introduced, now seem so natural as to have been inevitable. Indeed, it is difficult to fully grasp the influence his vast contributions to modern mathematics have subsequently had on new generations of mathematicians.Many of the groundbreaking contributions in these volumes contain material that is now considered foundational to the subject. Topics addressed by these top-notch contributors match the breadth of Grothendieck's own interests, including: functional analysis, algebraic geometry, algebraic topology, number theory, representation theory, K-theory, category theory, and homological algebra.CONTRIBUTORS to Volume II: P. Cartier; C. Contou-Carrère; P. Deligne; T. Ekedahl; G. Faltings; J.-M. Fontaine; H. Hamm; Y. Ihara; L. Illusie; M. Kashiwara; V.A. Kolyvagin; R. Langlands; Lé D.T.; D. Shelstad; and A. Voros.

This volume contains lectures delivered at the International Conference Operator Theory and its Applications in Mathematical Physics (OTAMP 2004), held at the Mathematical Research and Conference Center in Bedlewo near Poznan, Poland. The idea behind these lectures was to present interesting ramifications of operator methods in current research of mathematical physics.

The ISAAC Group in Pseudo-Differential Operators (IGPDO) met at the Fifth ISAAC Congress held at Universita di Catania in Italy in July, 2005. This volume consists of papers based on lectures given at the special session on pseudodifferential operators and invited papers that bear on the themes of IGPDO. Nineteen peer-reviewed papers represent modern trends in pseudo-differential operators. Diverse topics related to pseudo-differential operators are covered.

This volume! aims at introducing some basic ideas for studying approxima- tion processes and, more generally, discrete processes. The study of discrete processes, which has grown together with the study of infinitesimal calcu- lus, has become more and more relevant with the use of computers. The volume is suitably divided in two parts. In the first part we illustrate the numerical systems of reals, of integers as a subset of the reals, and of complex numbers. In this context we intro- duce, in Chapter 2, the notion of sequence which invites also a rethinking of the notions of limit and continuity2 in terms of discrete processes; then, in Chapter 3, we discuss some elements of combinatorial calculus and the mathematical notion of infinity. In Chapter 4 we introduce complex num- bers and illustrate some of their applications to elementary geometry; in Chapter 5 we prove the fundamental theorem of algebra and present some of the elementary properties of polynomials and rational functions, and of finite sums of harmonic motions. In the second part we deal with discrete processes, first with the process of infinite summation, in the numerical case, i.e., in the case of numerical series in Chapter 6, and in the case of power series in Chapter 7. The last chapter provides an introduction to discrete dynamical systems; it should be regarded as an invitation to further study.

This is the ?rst in a series of three volumes dedicated to the lecture notes of the Summer School "e;Open Quantum Systems"e; which took place at the Institut Fourier in Grenoble from June 16th to July 4th 2003. The contributions presented in these volumes are revised and expanded versions of the notes provided to the students during the School. Closed vs. Open Systems By de?nition, the time evolution of a closed physical systemS is deterministic. It is usually described by a differential equation x ? = X(x ) on the manifold M of t t possible con?gurations of the system. If the initial con?guration x ? M is known 0 then the solution of the corresponding initial value problem yields the con?guration x at any future time t. This applies to classical as well as to quantum systems. In the t classical case M is the phase space of the system and x describes the positions and t velocities of the various components (or degrees of freedom) ofS at time t. Inthe quantum case, according to the orthodox interpretation of quantum mechanics, M is a Hilbert space and x a unit vector - the wave function - describing the quantum t state of the system at time t. In both cases the knowledge of the state x allows t to predict the result of any measurement made onS at time t.